Gravity Regge Convergence Registry Riemann Convergence Faithful

The faithful projection

Regge calculus is a way of approximating curved spacetime by flat triangular pieces, the way a geodesic dome approximates a sphere. A central question is whether these piecewise-flat approximations converge to the smooth geometry they are meant to represent as the pieces get finer. In the machine-checked library of formal theorems, four external convergence results are collected into one named structure, the Regge convergence registry, a single object that bundles them together with a note on where each came from.

The declaration riemann_convergence_faithful is a theorem about one of those four inputs. It states that the registry's field for Riemann convergence, the property that holonomy around small loops approaches the smooth curvature, yields a proof of the exact original proposition when projected out. The theorem is a round-trip guarantee: you can put the original convergence result into the registry, take it back out, and you have the same proof you started with. The library proves this for all four fields, and the construction and projection operations are shown to be exact inverses.

What the theorem does not do is prove that Regge calculus converges. The Riemann convergence input is an external hypothesis, not a derived result. Its status is axiomatized, meaning the library takes it as given from outside mathematics. The theorem only establishes that the registry faithfully repackages this external input without weakening or altering it. No new convergence content is created, and no physical claim about gravity is added.

The value of the declaration is organizational. It lets the library treat four external results as one coherent bundle while preserving the ability to recover each original proof exactly. This matters for the framework's larger project of building a unified derivation of physics from recognition costs, because it keeps the boundary between what is proved and what is assumed clean and auditable.

/-- The `riemann_convergence` field of a registry is a proof of the original
`regge_riemann_convergence_axiom` proposition. -/
theorem riemann_convergence_faithful (r : ReggeConvergenceRegistry) :
    regge_riemann_convergence_axiom := r.riemann_convergence

/-- A registry that consolidates the four external convergence Propositions
from `NonlinearConvergence` into a single named structure.

Each field is typed by the **exact** original proposition, so this is a
faithful repackaging, not a weakening.  The `provenance` field documents
each input as an external-mathematics result with a status tag. -/
structure ReggeConvergenceRegistry where
  /-- CMS Theorem 5.1 curvature-measure bound (Cheeger–Müller–Schrader 1984). -/
  cms_measure_bound : cms_theorem_5_1_measure_bound
  /-- Special-purpose `O(a²)` action-convergence hypothesis. -/
  special_quadratic : special_quadratic_regge_to_eh_convergence_hypothesis
  /-- Regge Ricci-scalar convergence axiom. -/
  ricci_convergence : regge_ricci_convergence_axiom
  /-- Regge Riemann / holonomy convergence axiom. -/
  riemann_convergence : regge_riemann_convergence_axiom
  /-- Documentation of each external-math result with a status tag. -/
  provenance : List String

What this page does not claim

This answer does not claim that Regge calculus convergence is proved. This answer does not claim the registry creates new mathematical content.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/ReggeConvergenceRegistry.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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